Below is a picture of Devil’s Post pile, near Mammoth Lakes, California. These posts are cooled lava (called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns. However, variations in cooling caused some columns to either not be perfect or pentagonal.

First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each angle be? After completing this Concept you'll be able to answer questions like these.

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Guidance

Recall that interior angles are the angles inside a closed figure with straight sides. As you can see in the images below, a polygon has the same number of interior angles as it does sides.

A diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that the sum of the angles in a triangle is 180∘. What about other polygons?

Investigation: Polygon Sum Formula

Tools Needed: paper, pencil, ruler, colored pencils (optional)

1. Draw a quadrilateral, pentagon, and hexagon.

2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.

Make sure none of the triangles overlap.

3. Make a table with the information below.

Name of Polygon

Number of Sides

Number of △s from one vertex

(Column 3) × (∘ in a △)

Total Number of Degrees

Quadrilateral

4

2

2×180∘

360∘

Pentagon

5

3

3×180∘

540∘

Hexagon

6

4

4×180∘

720∘

4. Do you see a pattern? Notice that the total number of degrees goes up by 180∘. So, if the number sides is n, then the number of triangles from one vertex is n−2. Therefore, the formula would be (n−2)×180∘.

Polygon Sum Formula: For any n−gon, the sum of the interior angles is (n−2)×180∘.

A regular polygon is a polygon where all sides are congruent and all interior angles are congruent.

Regular Polygon Formula: For any equiangularn−gon, the measure of each angle is (n−2)×180∘n.

Example A

Find the sum of the interior angles of an octagon.

Use the Polygon Sum Formula and set n=8.

(8−2)×180∘=6×180∘=1080∘

Example B

The sum of the interior angles of a polygon is 1980∘. How many sides does this polygon have?